Lesson 17 MAT 1372 Statistics with Probability Sampling

Lesson #17 MAT 1372 Statistics with Probability

Sampling Distribution of Sample Means Population with μ, σ Sample 1 Sample 5 Sample 2 The sampling distribution consists of the values of the sample means, © 2012 Pearson Education, Inc. All rights reserved. 2 of 105

Properties of Sampling Distributions of Sample Means 1. The mean of the sample means, population mean μ. , is equal to the 2. The standard deviation of the sample means, , is equal to the population standard deviation, σ, divided by the square root of the sample size, n. • Called the standard error of the mean. © 2012 Pearson Education, Inc. All rights reserved. 3 of 105

Example: Sampling Distribution of Sample Means The population values {1, 3, 5, 7} are written on slips of paper and put in a box. Two slips of paper are randomly selected, with replacement. a. Find the mean, variance, and standard deviation of the population. Solution: © 2012 Pearson Education, Inc. All rights reserved. 4 of 105

Example: Sampling Distribution of Sample Means b. Graph the probability histogram for the population values. Solution: Probability Histogram of Population of x P(x) 0. 25 Probability All values have the same probability of being selected (uniform distribution) x 1 3 5 7 Population values © 2012 Pearson Education, Inc. All rights reserved. 5 of 105

Example: Sampling Distribution of Sample Means c. List all the possible samples of size n = 2 and calculate the mean of each sample. Solution: Sample 1, 1 1, 3 1, 5 1, 7 3, 1 3, 3 3, 5 3, 7 1 2 3 4 5 © 2012 Pearson Education, Inc. All rights reserved. Sample 5, 1 5, 3 5, 5 5, 7 7, 1 7, 3 7, 5 7, 7 3 4 5 6 7 These means form the sampling distribution of sample means. 6 of 105

Example: Sampling Distribution of Sample Means d. Construct the probability distribution of the sample means. Solution: f Probability 1 2 3 0. 0625 0. 1250 0. 1875 4 5 6 7 4 3 2 1 0. 2500 0. 1875 0. 1250 0. 0625 © 2012 Pearson Education, Inc. All rights reserved. 7 of 105

Example: Sampling Distribution of Sample Means e. Find the mean, variance, and standard deviation of the sampling distribution of the sample means. Solution: The mean, variance, and standard deviation of the 16 sample means are: These results satisfy the properties of sampling distributions of sample means. © 2012 Pearson Education, Inc. All rights reserved. 8 of 105

Example: Sampling Distribution of Sample Means f. Graph the probability histogram for the sampling distribution of the sample means. Solution: P(x) Probability 0. 25 Probability Histogram of Sampling Distribution of 0. 20 0. 15 0. 10 0. 05 2 3 4 5 6 7 The shape of the graph is symmetric and bell shaped. It approximates a normal distribution. Sample mean © 2012 Pearson Education, Inc. All rights reserved. 9 of 105